Bullseye illusion

In Figures 1 to 4, the green circle appears larger than the blue ring. In reality, however, both areas have the same area.

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  Figure 1
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  Figure 2
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  Figure 3
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  Figure 4

The blue ring is formed by the difference in area between a larger circle with radius c and a smaller circle with radius b.

Let the green circle have radius a. Then the condition for the equality of the two areas is:

${\displaystyle \pi a^{2}=\pi c^{2}-\pi b^{2}}$, i.e.,

${\displaystyle a={\sqrt {c^{2}-b^{2}}}}$.

For the special case ${\displaystyle a=b}$ (Figure 4), the condition for the equality of the two areas is:

${\displaystyle \pi b^{2}=\pi c^{2}-\pi b^{2}}$, i.e.,

${\displaystyle a=b={\frac {c}{\sqrt {2}}}}$.

In this case, the larger circle is completely decomposed into the smaller green circle and the blue ring of equal area.^[1][2]

• Bullseye illusion Wolfram MathWorld
• Bullseye illusion (interactive demonstration) Wolfram Demonstrations Project
• Bullseye optical illusion from math-fail.com